Authoritative might sound like it has one syllable too many, but it is the standard form of the adjective meaning having or arising from authority. Authoritive is arguably more logical—the root noun is authority, not authoritaty—but English is not always logical with these things, and authoritive only rarely appears in edited writing. In a Google Ngram graphing the words’ use over the past two centuries, the shorter form barely registers against the longer one. And although it is possible to find a few scattered instances of authoritive in texts from past centuries, authoritative has been the prevalent form since it entered English around 1600.
English is not only often illogical with these things, but also inconsistent. For instance, while argumentative (which is preferred over argumentive) also takes that extra syllable, preventive (preferred over preventative) does not. The reason the -ative suffix is often favored over -tive is that -ative more closely resembles suffixes used in French and Latin, and these languages have had strong influence on English at various stages. Words formed more recently are more likely to have the -tive suffix because brevity is now more important to us than etymological considerations.
Authoritive is rare, but examples are easily found on the web—for example:
Young people had given up wearing hats because they felt they were too authoritive, too conforming. [Irish Independent]
Leaving Emperor Visari defeated, two of Helghan’s authoritive figures are in a battle amongst each other to take over Helghan. [Examiner]
But the more conventional authoritative, as used in the following examples, is far more common and is considered the correct form:
Apparently yes, according to an authoritative group that specializes in separating myth from fact when it comes to medical treatments. [Washington Post]
The 26-page missive is seen as an authoritative guide to the state of the world’s biggest economy. [BBC News]
The survey is considered the most authoritative measure of sovereign risk for mining nations. [The Australian]